A125281
E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.
Original entry on oeis.org
1, 1, 3, 16, 149, 2316, 59047, 2429554, 159549945, 16557985432, 2693862309131, 682199144788734, 267277518618047797, 161130714885281760100, 148762112860064623199295, 209444428223095096806228346, 447998198975235291015396393713, 1450973400598977755884988875863216
Offset: 0
A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2316*x^5/5! +...
where
A(x) = 1 + x*A(x) + x^2*A(2*x)/2! + x^3*A(3*x)/3! + x^4*A(4*x)/4! + x^5*A(5*x)/5! +...
which leads to the recurrence illustrated by:
a(3) = 1*3^0*(1) + 3*2^1*(1) + 3*1^2*(3) = 16;
a(4) = 1*4^0*(1) + 4*3^1*(1) + 6*2^2*(3) + 4*1^3*(16) = 149;
a(5) = 1*5^0*(1) + 5*4^1*(1) + 10*3^2*(3) + 10*2^3*(16) + 5*1^4*(149) = 2316.
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *)
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{a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)*(n-k)^k*a(k)))}
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{a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,x^k/k!*subst(A,x,k*x)+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A218682
E.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n/n! * A(n*x)^n.
Original entry on oeis.org
1, 1, 6, 93, 2944, 167685, 16037376, 2481455137, 609371157312, 235171042752105, 141778378915235200, 132871933962627534741, 192678752056300896500544, 430404642833695770472870573, 1474859949355240010986735351872, 7723091241704594423130951106689225
Offset: 0
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 93*x^3/3! + 2944*x^4/4! + 167685*x^5/5! +...
where
A(x) = 1 + x*A(x) + 2^2*x^2*A(2*x)^2/2! + 3^3*x^3*A(3*x)^3/3! + 4^4*x^4*A(4*x)^4/4! +...
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m = 16; A[_] = 0;
Do[A[x_] = Sum[If[n == 0, 1, n^n x^n/n! A[n x]^n], {n, 0, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m - 1]! (* Jean-François Alcover, Oct 03 2019 *)
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{a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,k^k*x^k/k!*subst(A,x,k*x)^k+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A385585
G.f. A(x) satisfies A(x) = Sum_{k>=0} (k*x)^k * A(k*x).
Original entry on oeis.org
1, 1, 5, 40, 457, 7101, 148270, 4206121, 165267951, 9117777074, 709325010385, 77906424970811, 12109278363587036, 2670187179684919761, 836451775445907622685, 372646977140600929476104, 236390047765997660237447061, 213719117789650238860723125601
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[(n-k)^n * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 04 2025 *)
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (i-j)^i*v[j+1])); v;
Showing 1-3 of 3 results.